Elemente der Euklidischen Geometrie. Göttingen, Wintersemester 1898/99. Mechanically reproduced manuscript in a professional copyist’s hand.

Göttingen: 1899.

First printing, extremely rare, of Hilbert’s classic lectures on the foundations of geometry, delivered during the 1898-99 winter semester at Göttingen. It precedes the revised and expanded published version, retitled “Grundlagen der Geometrie,” that was included in the Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen (1899). This prepublication version, reproduced from a professionally prepared manuscript, was privately printed in an edition of 70 copies under the direction of Hans von Schaper, one of Hilbert’s doctoral students; as noted in von Schaper’s introduction (p. [ii]), it was intended primarily for the benefit of those who attended Hilbert’s mathematical lectures. Hilbert’s work on the foundations of geometry represents the most successful attempt to establish a complete set of axioms from which Euclidean geometry can be derived. “Hilbert’s idea was to begin with three undefined terms, point, straight line, and plane, and to define their mutual relations by means of the axioms . . . The importance of Hilbert’s work lay not so much in his answering the various objections to parts of Euclid’s deductive scheme, but in reinforcing the notion that any mathematical field must begin with undefined terms and axioms specifying the relationships among the terms . . . There were many axiom schemes developed in the late 19th century to clarify various areas of mathematics. Hilbert’s work can be considered the culmination of this process, because he was able to take the oldest such scheme and show that, with a bit of tinkering, it had stood the test of time. Thus, the mathematical ideas of Euclid and Aristotle were reconfirmed at the end of the 19th century as still the model for pure mathematics” (Katz, History of Mathematics, pp. 719-721). This initial printing of Hilbert’s Elemente der Euklidischen Geometrie is what is known to Hilbert scholars as an ‘Ausarbeitung,’ a more polished written version of a series of lectures Hilbert had previously given at Göttingen’s Mathematisches Institut. “As a rule, the Ausarbeitungen came about as follows. Hilbert would ask one of his assistants or collaborators (generally his own graduate students, but sometimes advanced students, doctoral students of other professors, or other collaborators) to take notes of his lectures and then work them up into a polished finished product. Hilbert himself would generally supervise this process closely, first discussing the lectures in advance with the Ausarbeiter, and later correcting the written product, usually before its mimeograph reproduction . . . Somewhere between ten and twenty of these official Ausarbeitungen were deposited during Hilbert’s lifetime in the Lesezimmer [reading room] of the Mathematical Institute where they were freely accessible to 28 students” (Hallett & Majer, p. xiii). Hilbert’s original manuscript notes for the present lecture have been preserved, but the lecture’s Ausarbeitung presents a much more complete view of his conception of the foundations of geometry: “it often contains fully worked out versions of proofs which are only sketched in the notes . . . and without question gives a more complete picture of Hilbert’s intentions” (Hallett & Majer, p. 189). Occupying a middle position between Hilbert’s original lecture and the final published version, Elemente der EuklidischenGeometrie provides valuable insight into the evolution of Hilbert’s mathematical thought. Five copies located in US (East Carolina U., Dartmouth, Brown, Notre Dame, Washington State U.). Not on RBH.

Provenance: Karl Sigismund Hilbert (no relation), one of Hilbert’s doctoral students, who obtained his Ph.D. in mathematics from Göttingen in 1900 with a dissertation entitled ‘Das Allgemeine Quadratische Reciprocitatsgesetz in ausgewahlten Kreiskorpern der 2 Einheitswurzeln.’

“Hilbert’s first lectures on geometry dealt with Projektive Geometrie (1891). They dealt with the properties that are invariant under projections. Then came a manuscript on non-Euclidean geometry, axiomatically formulated; Die Grundlagen der Geometrie [‘GG’] of 1894. The third source relates to an Easter vacation course in 1898, Ueber den Begriff des Unendlichen [‘FK’]; it seems to form the kernel of the later book. From it emerged the detailed manuscript on Euclidean geometry, written in the winter semester 1898–1899: Grundlagen der Euklidischen Geometrie [‘EG’]. Finally, the elaboration Elemente der Euklidischen Geometrie (‘SG’), was prepared by Hilbert’s assistant Hans von Schaper from the preceding lectures in March 1899; and from that text Hilbert developed the book, which was published in June 1899” (Toepell, p. 713).

“At Easter in 1895 Hilbert accepted the chair at Göttingen and up to 1897 he concerned himself principally with number theory. So his concern with the foundations of geometry rested for more than three years, until he was inspired to take it up anew by a letter of 30 January 1898 sent by Friedrich Schur to Klein. Hilbert wrote in March 1898 to Hurwitz: ‘This letter, which [Artur] Schoenflies introduced to us in a lecture to the mathematical society, has given me the inspiration to take up again my old ideas about the foundations of Euclidean geometry. It is remarkable how many new things can be discovered in this field’.

“The manuscript (FK) of an Easter vacation course of 1898, Ueber den Begriff des Un- endlichen (‘On the concept of infinity’), covers just 27 pages; but it forms the nucleus of the Festschrift. As we see from the introduction to this course, the contact with teachers and school-mathematics was Hilbert’s personal request. He addressed himself especially to the teachers as ‘the most competent collaborators’; perhaps they who especially stimulated him to study the foundations of geometry. A reviewer even postulated that the Grundlagen der Geometrie should be used as a textbook in school geometry, like a new Euclid. Indeed, it became a fundamental textbook in university geometry in the 20th century.

“In this course Hilbert introduced his audience to the most up-to-date research questions. For the first time he constructed the axioms in what was subsequently to be their usual sequence. Then he directed the teachers to practical problems: the geometrical constructions based on the theorems of congruence. For example, the constructibility of the intersecting point of two circles required an axiom of continuity, whose independence was subsequently examined. Also he asked for the first time, which axioms were dispensable if one assumed Desargues’s and Pascal’s theorems in place of some axioms that were used to prove these theorems.

“In this manuscript the arrangement of the later Festschrift is already apparent: axioms, proofs of independences, segment arithmetic, Desargues’s theorem, Pascal’s theorem, and problems concerning constructibility. We can also trace how Hilbert developed his ideas in two directions: to avoid assumptions of continuity, and to construct plane geometry independent of spatial assumptions. Once Hilbert’s basic concept had been established, a number of individual problems came into focus on which he now worked intensively. That led him to the careful system in the Festschrift.

“In the winter semester of 1898–1899 we read in the announcements of lectures in Göttingen: ‘Elemente der Euklidischen Geometrie: Prof. Hilbert, Montag und Dienstag 8–9 Uhr, privatim’. So, two hours per week. Hilbert began: ‘Concerning the content of the lectures, we shall study the theorems of elementary geometry, which we all learned at school: the theory of parallels, the theorems of congruence, the equality of polygons, the theorems about the circle etc. in the plane and the space’.

“The manuscript EG contains an exhaustive discussion of those areas that were mostly treated in brief in the vacation course. The logical meaning of the axioms was studied by construction of arithmetical models. Amongst these were proofs of independence for axioms of the first two groups. In accordance to the theme of the lectures, Hilbert examined in detail the studies of congruence that were possible without using continuity. Much of this was omitted in the Festschrift, including (unusually in his writings) an historical survey of the parallel axiom that follows, then the detailed presentation of a non-Euclidean geometry and the introduction of ideal (infinite) elements.

“Comparing this lecture with that of 1894, it is plausible when Klein remarked of the Festschrift that ‘compared with earlier studies its main object is to state the importance of the axioms of continuity’. Freudenthal asserted that ‘The so-called axioms of continuity are introduced by Hilbert to show that actually they are dispensable’. Because of this important result, Hilbert introduced them at the end.

“In March 1899 Hilbert’s assistant von Schaper had elaborated these lectures as the text Elemente der Euklidischen Geometrie (SG). It contains numerous remarks, motivations and examples, which were omitted in the concise presentation of the Festschrift.

“Here Hilbert began with the fundamental concepts. He did not explain it as in the lectures ‘es giebt ein System von Dingen, die wir Punkte nennen’ (‘there is a system of things, that we call points’), but formulated it with abstract rigour: ‘Zum Aufbau der Geometrie denken wir uns drei Systeme von Dingen, die wir Punkte, Geraden und Ebenen nennen’. In the Festschrift he omitted even the words ‘zum Aufbau’ and ‘uns’ and wrote: ‘Wir denken drei verschiedene System von Dingen’; Leo Unger translated this as ‘Consider three distinct sets of objects’ in the second English edition of 1971.

‘With these lion-claws the navel-string between reality and geometry is cut through’ [Freudenthal]. Geometry seems to awake to its own existence, independent of any physical reality. Some months before Hilbert had still seen the axioms as ‘very simple [. . . ] original facts’ (SG), whose validity is experimentally provable in nature.

“What Hilbert formulated may have been new in Germany, but it was ‘in the air’. Already seven years earlier Fano wrote: ‘At the basis of our study we put some variety of entities of some nature; entities that we shall call, for brevity, points, but independently, well agreed, of their actual nature’. In the two years following (1892–1894) Fano had been in Göttingen.

“Concerning the further development it is interesting that the elaboration included studies of some theorems of Legendre that Hilbert published only thirty years later in the seventh edition (art. 10, 39–45). An eight-page introduction to projective geometry by means of ideal elements was also omitted by Hilbert from the Festschrift

“Hilbert’s aim from the outset seems to be the algebraisation of geometry. In 1894 he still was satisfied with the introduction of coordinates by means of the Möbius grid. At the end he established that it must also be possible to calculate with these numbers ascribed to geometrical objects. Hence the algebraic laws of fields (‘Körpergesetze’) were required.

“While Pasch had spoken of primitive propositions ‘directly based on observation’, from which he derived all the remaining theorems, for Hilbert the relations between the objects of intuition provide the starting point, as in his manuscript of 1894. Having perceived both the starting-point and the aim, it remains only to find the way. Like Euclid, Hilbert proceeds axiomatically. Here the question arises, which axioms are required?

“While the axioms of incidence were largely clear, those of order were somewhat problematical. Hence, because of infinite elements, difficulties attended projective geometry. Proceeding to Euclidean geometry, the concept of congruence could be introduced without hesitation. But then appeared the problem of the intersection theorems, of the axioms of parallels, of the Archimedian axiom, and of continuity—all aspects that are not mutually independent. As the details demonstrated, it was not easy for him to find a suitable way through this maze of axioms. (ibid, pp. 716-718).

The Festschrift led to the world-wide reputation of Hilbert. The first written congratulations came from Minkowski, Hurwitz and Aurel Voss. One of the first public reactions to the epistemological background can be found in a lecture by Otto Hölder a month after publication, on 22 July 1899. The philosophical discussion about the nature of axioms with Gottlob Frege led to Hilbert’s decisive article ‘Ueber die Grundlagen der Logik und Arithmetik’ in 1905, a philosophical Programmschrift

“The book had some consequences for physics, which gained Hilbert’s attention from the 1900s onwards. For during the 1910s there was a remarkable connection between Ein- stein and Hilbert. Seeking the roots and sources of Einstein’s ideas in geometry, we are led back to the time of his being a student of Minkowski at the Polytechnical High School in Zurich—especially in 1899, when he had read the proof-sheets of Hilbert’s work. According to his letters we may assume that he understood the immense power of this book and it should not take too long for initiating Einstein as well. Maybe Einstein learnt from Hilbert to free himself from empirical restrictions in geometry. This led 12 years later to the idea to think of spatial curvature not only in a Euclidean or non-Euclidean form but also in a form emerging out of gravitational forces.

“It is not at all obvious that the conception of the Grundlagen der Geometrie emerged from a vacation course for teachers; the significance of intuition seems to be entirely subordinate. Also in his further publications Hilbert argued as a rule for the axiomatic method. Hence he was frequently seen as a formalist. However, he never once used ‘formalism’ to characterise his philosophical position, and his manuscripts and letters show his intense concern with intuition and its significance for geometry. Regarding his attitude in later years we see how little Hilbert freed himself from intuition. He perceived that the consistency of his axiomatic system depends after all on what it means.

“One hundred years after Hilbert’s first edition, the famous geometer Gian-Carlo Rota wrote:

‘Today, synthetic geometry is still the downside. For today’s students of algebraic geometry, points, lines and surfaces are a manner of speaking, shorthand terms for algebraic concepts. But the call to reality is making itself felt. Computer scientists have shown us how little we know about solid angles, how much we need to know about polyhedra. The visual geometry of Euclid, Desargues, Ludwig Schläfli and Eugenio Cremona is about to make a triumphal comeback. Geometers of today will be well advised to recover their bearings by reading Hilbert’s Grundlagen der Geometrie’” (ibid., p. 722)

The following materials are offered with the present copy of Elemente der Euklidischen Geometrie:

  • Hilbert, Karl Sigismund. Das allgemeine quadratische Reciprocitätsgesetz in ausgewählten Kreiskörpern der 2hten Einheitswurzeln. Inaugural-Dissertation . . . Göttingen: Dieterich’sen Universitäts-Buchdruckerei, 1900. Pp. iv, 72, [2]. Original printed wrappers. Front wrapper inscribed by the author: “Herrn Professor Dr. Hälder überreicht vom Verf.”
  • Hilbert, Karl Sigismund. Die Philosophie der geistigen Schöpfung. Weimar: Panses Verlag, 1930. Pp. 54. Original printed wrappers. Title-page inscribed by the author: “Frau Käte Rautenstrauck, Trier überricht vom Verfasser (Dez. 1930).”
  • Toepell, Michael. On the origins of David Hilbert’s “Grundlagen der Geometrie.” Offprint from Archive for the History of Exact Sciences 35 (1986). Pp. 329-344. Without wrappers.

Hallett & Majer, David Hilbert’s Lectures on the Foundations of Geometry 1891-1902, pp. xi-xxviii; 186-196; text of the Elemente der Euklidischen Geometrie reproduced on pp. 302-395. Toepell, ‘David Hilbert, Grundlagen der Geometrie, First Edition (1899),’ Chapter 55 in Landmark Writing in Western Mathematics 1640-1940, Grattan-Guinness (ed.), 2005.



Small 4to (214 x 161 mm), pp. [4], 175, with numerous diagrams in text (first two leaves repaired, light toning, a few margins closely trimmed). Contemporary cloth, gilt-lettered spine (light wear). A very good copy.

Item #6076

Price: $12,500.00